Stochastic Optimization for Lake Eutrophication Management

Man-made (or artificial) eutrophication has been considered as one of the most serious water quality problems of lakes during the last 25-40 years. Increasing discharges of domestic and industrial waste water and the intensive use of crop fertilizers -- all leading to growing nutrient oads of the recipients -- can be mentioned among the major causes of this undesirable phenomenon. The typical symptoms of eutrophication are among others sudden algal blooms, water coloration, floating water plants and debris, excreation of toxic substances causing taste and odor problems of drinking water and fish kills. These symptoms can easily result in limitations of water use for domestic, agricultural, industrial or recreational purposes.

One of the major features of artificial eutrophication is that although the consequences appear within the lake, the cause -- the gradual increase of nutrients (various phosphorous and nitrogen compounds) reaching the lake -- and most of the possible control measures lie in the region. Consequently, eutrophication management requires analysis of complex interactions between the water body and its surrounding region. In the lake, different biological, chemical and hydrophysical processes -- all being time and space dependent, furthermore non-linear -- are important, while in the region one must take into account human activities generating nutrient, residuals and control measures determining that portion of the emission which reaches the water body.

Eutrophication management requires a sound understanding of all these processes and activities, which belong to diverse disciplines. Additionally, various uncertainties and stochastic features of the problem have to be also taken into account -- the estimation of loads from infrequent observations and the dependence of water quality on hydrologic and meteorologic factors. The fact that we are dealing with a stochastic environment is especially important for shallow lakes, primarily due to the absence of thermal stratification that predicates a much more definite response to randomness as would be the case for deep lakes.

The paper presents an approach that combines descriptive, simulation and management optimization models. The roles of each of these models is discussed in Section 2. The derivation of the aggregated lake and planning type nutrient load models to be used in the management model is the subject of Section 3. Alternative management models are formulated in Section 4. Two of them were implemented: a ‘true’ stochastic model (which uses as the starting point of the iterative solution procedure the corresponding deterministic model) and a linear programming approach capturing stochastic features of the problem through a linearized expectation-variance model. A brief survey of the solution approach to the stochastic problem is presented in Section 5. In Section 6 the methods are applied to Lake Balaton and the results compared.

This chapter is based on the original technical report Somly´ody and Wets (1985) parts of which appeared in Somlyody and Wets (1988).